A good way to start a hard math problem is by playing around with it. Poking at it. Trying things to see what does and doesn’t work and to figure out why that is.
For a certain type of math problems, it’s helpful to just guess and then analyze why that guess isn’t right.
I don’t know if you’ve ever played the game Mastermind, but Mastermind is exactly this idea. One player hides 4 pin colors, and the second player has to guess what the colors are and where they’re placed. Each turn player two is given information on how much ze got wrong and how wrong it was. The only way to start is with a completely blind guess. If you guess right on the first try, the game isn’t very much fun. That means you won by luck and not by being able to actually play the game.
DC1 had never heard of such a thing before we got the Hard Math book. Ze was completely and totally frustrated by the first challenge problem (What is the largest possible answer to 782 + ABC =? [with carrying 1s above and above/left of the 7]?) because ze thought ze should just be able to do a math problem. Even hard math problems were hard because ze was prone to make mistakes, and all one had to do was not make mistakes. This idea that you have to learn about the problem first and maybe try a few things out was completely foreign to hir.
Life is like that too. You can plan and plot and analyze situations, but sometimes that takes more time (and provides less information) than just doing and seeing what happens. Sometimes you get what you want on the first try, but more often, you get clear information on what you need to do better and how and why.
Sometimes you have to fail before succeeding, and it’s the failure(s) itself that is instrumental to your eventual success.
Grumpy Rumblings (of the formerly untenured) 
March 25, 2014 at 5:51 am
That’s foreign language education in a nutshell.
March 25, 2014 at 6:13 am
Yup, which is one of the big problems of testing pre the internet era. You don’t find out if you’re right or wrong on a problem for a long time, and being wrong is bad — so there’s no learning going on. Whereas an online assessment could tell you instantly if you’re right or wrong and then adjust, so testing can actually become learning…
March 25, 2014 at 6:33 am
I’m not sure I agree 100%. Most of the time the teacher would have us review the tests thoroughly once we got our graded papers back, so we would walk through different ways to answer, how to get the answer, etc. This isn’t true for standardized assessments, where the answers are guarded secrets, but we worked through released (old versions) practice tests in those cases whenever possible. It wasn’t instantaneous, necessarily, but we still did it.
March 25, 2014 at 6:38 am
Wow, Mastermind! I haven’t played that in YEARS but it was sooooo “the thing” in elementary school before/after school care.
I feel ze’s pain, though. I much rather like to know WHY I know or don’t know something. I remember writhing in agony on the couch while doing math homework. I wouldn’t NOT do it, because I wanted to know what the answer was. But it was sooo frustrating. (I like math now, even though I got an MA in history….)
March 25, 2014 at 7:56 am
My sister and I LOVED Mastermind as kids! Thanks for reminding me about it- I’ll have to get it for my kids soon.
I think there are lots of areas where being willing to be wrong and learn from it is important. I use this technique a lot when I program. Of course, that means I need really good test cases…
March 25, 2014 at 8:10 am
I spent this morning trying to break my student’s survey. They’d already done that stress-testing and done it well, so they’d fixed all the potential problems I could think of. :)
Making mistakes and learning from them also helps folks remember potential problem spots.
March 26, 2014 at 10:19 pm
YES re learning from the mistakes as a way of remembering problem spots. I am trying to do more hands-on exercises in my library instruction sessions (I’m the history librarian) for exactly this reason: if they do the searching or whatever and get whatever nudge or help they need to get past a problem spot, they’ve got a better sense of what the *whole* process feels like, and, hopefully, how to get unstuck when they get to the sticky part. I hate having to do “assessment” for these classes where I judge who got the answer “right” and who got it “wrong,” because I’d a lot rather have the ones getting it “wrong” be able to flag me down and be walked through it.
Mastermind is a great example. Haven’t thought of that game for years.
March 25, 2014 at 9:50 am
I love mastermind. I’ve used it in my classroom from time to time. Does DC1 play?
Teaching trial and error is so challenging. I find this hard to do in school, where we are time limited, and students have to be learning. I do teach an outreach enrichment science class, and that’s all trial and error. Kids ask how to do something, and I usually counter with “what do you think?” or “why don’t you try that?” unless what they suggest is dangerous. It is a lot of fun to watch the students think and untangle complicated problems. Some of these kids are even the ones who are turned off in school.
One of my tasks for this summer is to work on making my classes more like the outreach. It’s tricky, and I can only do little moments here or there right now, but I want to gradually increase my inquiry-based learning. Of course, a key element is student interest/buy-in, and I can’t yet figure out how to get that.
March 25, 2014 at 9:59 am
DC1 has not in the past been a fan of mastermind (or checkers or anything that ze can’t beat mommy or daddy at on the first try). It’s that perfectionism streak. Ze has gotten a LOT better at not getting things right the first time in video games and sounding terrible the first time through a piano piece, and so on. So, baby steps. We’ll have to try it again one of these days if we can pry hir away from Pokemon and Mario.
March 25, 2014 at 11:55 am
I have not played Mastermind, but your post made me think of the game Clue, which I just taught to a group of elementary school children last week. The game is based on making uninformed guesses, learning from those guesses, and then solving the “crime” by process of elimination. It was interesting to watch the children (ages 8-11, some from a special needs class) play the game.
March 25, 2014 at 12:10 pm
definitely!
March 25, 2014 at 1:56 pm
I hadn’t thought about how many children’s games are based on the trial and error thing: Guess Who, Clue, Battleship…
March 25, 2014 at 2:11 pm
yeah! battleship!
March 25, 2014 at 6:36 pm
My kids seem to have an easier time with Battleship than with Clue–perhaps because the tracking of “guesses” is more visual and concrete in Battleship?
March 25, 2014 at 8:22 pm
This is true with figuring out what you like. Trying a bunch of stuff that you start off knowing nothing about is good. Sometimes you like things just because you have a crush on whoever introduced them to you; sometimes you find out new things about who you are.
March 28, 2014 at 10:19 pm
Oh, I so absolutely agree. My life has been set based on so many different plans. The ability to adjust those plans based on circumstances or “failures” has become one of my strongest talents based out of necessity. And in the long run things really have worked out for the best.
April 5, 2014 at 8:31 pm
The best part about math is after you walk away from a really hard problem, totally pissed off and frustrated because you can’t figure it out, Then you go to bed, and wake up the next morning knowing how to get the answer. And you’re right.